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Elastic, inelastic and super elastic collisions and generalized form of Newton’s Third Law
of motion
Ajay Sharma
Fundamental Physics Society His Mercy Enclave,
Post Box 107 GPO Shimla 171001 HP India
Email: [email protected]
In Newton’s Principia the Third Law of Motion states: ‘to every action there is always
opposed an equal reaction; or the mutual actions of two bodies upon each other are always
equal, and directed to contrary parts.’ The Law only refers to the ‘bodies’, not other factors.
The Law does not take into account the inherent characteristics, nature, compositions,
flexibility, rigidity, magnitude, distinctiveness of interacting bodies etc. The bodies may be of
steel, wood, rubber, cloth, wool, sponge, spring, typical plastic, porous material, specifically
fabricated material etc. For all such bodies if the action is same, then the reaction must be the
same. In the qualitative explanation given after the definition to the Law, Newton expressed
Action and Reaction in terms of push or pull (force) and velocity. It can be mathematically
understood that in elastic collisions (coefficient of restitution, e = 1) the Action and Reaction
are equal only under certain conditions. In numerous cases we find that action and reaction
are not equal as the Principia’s third law of motion as it does not take in account vital factors
like inherent nature and characteristics of interacting bodies. Thus, to take elusive factors
into account, the Third Law of Motion is generalized as: ‘To every action there is always an
opposed reaction, which may or may not be equal in magnitude, depending upon the inherent
characteristics of the process.’ Mathematically, Action =K Reaction, the value of K can be
equal to, less than, or greater than unity. The value of coefficient of proportionality K takes in
account the inherent characteristics of the process. It can be easily realised that the existing
observations justify the generalized form of the third law.
1.0 The Principia’s Third Law of Motion
The original form of the Third Law of Motion is:
To every action there is always opposed an equal reaction; or the mutual actions of two
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bodies upon each other are always equal, and directed to contrary parts. [1, 2]
Action = - Reaction
(1)
or Mutual action of one body = - Mutual reaction of other body
(1)
In Newton’s time it was beginning of science, thus physical quantities, units, dimensions were
not defined. Thus the terms action and reaction do not possess units and dimensions. However in
the qualitative explanation given after the definition to the law, Newton expressed action and
reaction in terms of push or pull (force) and velocity. Newton did not write eq. (1) in the
Principia.
Newton’s Original Explanation of the Third Law of Motion in the Principia
“Whatever draws or presses another is as much drawn or pressed by that other. If you press a
stone with your finger, the finger is also pressed by the stone. If a horse draws a stone tied to a
rope, the horse (if I may so say) will be equally drawn back towards the stone: for the distended
rope, by the same endeavour to relax or unbend itself, will draw the horse as much towards the
stone as it does the stone towards the horse, and will obstruct the progress of the one as much as
it advances that of the other.
If a body impinges upon another and by its force change the motion of the other, that body also
(because of the quality of, the mutual pressure) will undergo an equal change, in its own motion,
towards the contrary part. The changes made by these actions are equal, not in the velocities but
in the motions of bodies; that is to say, if the bodies are not hindered by any other impediments.
For, because the motions are equally changed, the changes of the velocities made towards
contrary parts are reciprocally proportional to the bodies. This law takes place also in
attractions, as will be proved in the next scholium.”
Thus in the short description of the law Newton did not give any equation
to measure or calculate the magnitudes of action and reaction. Thus Newton provided
conceptual, thematic and philosophical explanation of the law, not quantitative basis. Newton’s
law is applicable to all interacting bodies, action and reaction are always equal and opposite for
all interacting bodies. The definition of the law has no restrictions on its applications i.e. it is
universally applicable. Now law has been interpreted in those experiments, which are not
critically discussed so far. Thus it can be safely concluded that the law does not take in account
the significant factors such as inherent nature and characteristics of body. Consequently
Newton’s third law is generalized, so that the law becomes complete taking all factors in
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account.
1.1 The third law of motion blatantly neglects significant inherent nature and
characteristics of body.
The definition of the third law is such that it holds good under all circumstances as there are no
constraints on the inherent characteristics and nature of colliding bodies. There are no restrictions
on applicability of the law as in eq. (1). If the action is same for all bodies of sponge, spring and
rubber then reaction must be same, according to definition of the law. The first part of statement
simply says that action is always equal. But it does not say what action is? What is reaction?
How they are originated? However in second part the mutual impacts of bodies are mentioned as
action and reaction. In the explanation of the law after definition Newton expressed the action
and reaction in terms of force, push or pull, velocity etc. The most sensitive point is that the law
only talks about the ‘bodies’ and no other factors are considered at all. Thus the third law
establishes the universal equality between action and reaction.
Thus law does not consider the inherent characteristics, nature, composition,
flexibility, rigidity, magnitude, distinctiveness of interacting bodies. There is no factor which
accounts for the above significant factors. The bodies may be composed of steel, wood, rubber,
cloth, wool, sponge, spring, typical plastic, porous material, especially fabricated etc. For all
such bodies if the action is the same, then the reaction must be the same. Some bodies may have
inherent tendencies to restore original shape when deformed e.g. rubber or spring bodies.
Whereas others donot show any such tendency e.g. mud or flour balls. According to Newton’s
third law everybody experiences action and reaction and magnitudes of action and reaction are
equal and opposite universally. Furthermore, in collisions comparative size and point of impact,
of target and projectile, roughness of surfaces and resistive forces play significant roles. These
are very significant factors affecting the results and are taken in account via a coefficient of
proportionality in the generalized law. In the generalized or extended form of the law, a
coefficient of proportionality comes in equation which takes all elusive factors into account.
Further Newton’s law is completely silent about electrical and magnetic interactions between the
various bodies.
2.0 The various types of collisions and the Third Law of Motion.
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Newton had described the interactions of colliding bodies in third example of third law of
motion. Collisions in which both momentum and kinetic energy of the system are conserved are
called elastic collisions [3]. The coefficient of restitution or coefficient of resilience is unity (e =
1). Collisions in which the momentum of the system is conserved but kinetic energy is not
conserved are called inelastic collisions. Consider two bodies A (projectile) and B (target) of
masses M1 and M2 moving along the same straight line with speeds u1 and u2 respectively. The
bodies will collide only if u1 > u2.
v1 =
v2 =
The final speeds of bodies A and B are v1 and v2.
M 1  M 2 u1  2M 2u2
M 1  M 2 
M 2 M 1 u2  2M 1u1
M 1 M 2
(2)
(3)
If the target is at rest (u2 = 0), then Esq. (2-3) become
v1 =
M 1  M 2 u1
M 1  M 2 
(4)
2M 1u1
M 1 M 2
(5)
v2 =
The various sub-cases are discussed below.
(i) When M2 >> M1 i.e. target (body B) is very massive compared to the projectile (body A).
Thus
M1 - M2 = -M2,
M2 + M1 = M2
In this case the final speeds of the projectile and target can be calculated from eq.(4) and eq.(5).
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v1 =
 M 2u1
= -u1
M2
(6)
v2=0.
The negative sign in v1 indicates that direction of body A reverses after collision. Now action
and reaction are expressed in terms of speed:
Action (u1) = -Reaction (v1)
e=
v2 v1 u1
=
=1
u1  u 2 u1
(7)
As the coefficient of restitution is unity, the collision is elastic. Thus Newton’s Third Law of
Motion is justified. However these are ideal mathematical calculations as we have not considered
actual experimental characteristics at all. The composition of various bodies may be different
e.g. body B can be of cloth and body A of wood, body B can be an air filled ball and body A of
aluminum, body B may be an air filled football and body A of gold, etc. In addition, the point of
impact must be at the centre of the target; if not then the direction of recoil will be different. The
various and distinct experiments must be conducted for final conclusions to be drawn.
2.1 Mass of body B is 1000 times larger than that of body A
If we consider that the target is 1000 times more massive than the projectile i.e. M2 = 1000M1,
then
v1
= -0.998001998u1
(8)
(final velocity of projectile) or Reaction = - 0.998001998u1 (initial velocity of projectile) or
Action;
or
Action (u1) = -1.002002 Reaction (v1)
e=
2M 1u1  999M 1u1
v2 v1
=[
]/[u1-0] = 1
1001M 1
u1  u 2
1001M 1
Thus the collision is elastic. But the action is greater than the reaction. Hence conclusions
cannot be drawn from a single observation. The calculations can be conducted for other
parameters as well.
3.0 The striking of a rubber ball on a wall
(9)
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This case is equivalent to elastic collisions (M2 >> M1), as discussed above. Consider a boy
standing in front of a rigid concrete wall, holding a round rubber ball and a cloth ball in different
hands. Let the wall be 5m from the boy. Let the boy throws the rubber ball at the wall with a
force of Faction (1N, say). The ball strikes the wall and comes back to the boy, travelling 5m (say)
in time t. Let the action be force Faction (1N, say).
The wall pushes the ball backward towards the boy through a distance 5m in time t. Thus the
wall exerts a force (Freaction = 1N) in reaction. So,
Boy pushes the ball through 5m in time t = Ball rebounds from wall through 5m in time t
Action (Faction = 1N) = -Reaction (Freaction = 1N)
(10)
In this case action and reaction are equal; hence The Principia’s Third Law of Motion is justified
under these conditions.
3.1 The striking of balls of different compositions upon different walls
Let the boy throws the cloth ball (softer, flexible, stretchy or typical) at the wall with a force of
Faction (1N) on the wall, which is at distance of 5m. The ball strikes the wall in time t, and
rebounds to only distance of 2.5m in the same time t. The wall pushes the ball backward towards
the boy through a distance 2.5m in time t. Thus the wall exerts a force (Freaction = 0.5N) in
reaction. Now compare both the cases, i.e. rubber ball and cloth ball.
Action (Faction = 1N)  -Reaction (Freaction = 0.5N)
In this case, action and reaction are not equal, although opposite. There are
numerous cases like this. Also the wall can be a normal concrete wall, a concrete wall with 10cm
covering with cloth, a steel wall, a wall made of wooden planks, a wall of paper sheets,
cardboard, etc.
The perfectly elastic collision (e=1) is a rare event. We treat collisions between ivory or glass
balls nearly elastic [4]. Almost any collision we observe on a human scale will be inelastic
(e<1). Newton’s law does not account for the characteristics of interacting bodies. Momentum is
conserved in both types of collisions, but kinetic energy is "conserved" in only elastic collisions.
A bullet shot into a tree would be one case. The final kinetic energy is nearly zero and bullet
does not rebound. It is perfectly inelastic collision.
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In super elastic collision the potential energy is converted into kinetic energy so that the
total kinetic energy of the colliding objects is greater after the collision than before. Imagine a
massive spring of high spring constant being compressed with an extremely delicate device (i.e.
once touched will release the spring). Now we can envision that a collision between a slowly
moving particle and this device, would release the spring and the final kinetic energies of the
massive spring and the particle will be larger than their initial one, because the elastic potential
energy was converted into kinetic energy. This situation can also be visualized in the following
way.
Consider a specially and typically fabricated round rubber ball and wall (say metallic).
Then a ball is thrown with some force (action) from distance 1m and strikes the wall in 2s (u1=.
0.5m/s). After striking the wall the rubber ball rebounds to initial position at distance 1m in 1s
(v1 = 1m/s). Thus reaction is double than action. This is example of super elastic collisions. It can
be understood in various other way also. For example, nitrocellulose billiard balls can literally
explode at the point of impact. This situation can be realized in different ways. According to
third law of motion, action and reaction of interacting bodies are always equal i.e. universally
equal, but it is not illustrated in above observations. The above experiments or observations are
discussed in the literature but not in view of third law of motion.
The various results are given in Table I.
Table I: Comparison of action and reaction on rubber and cloth balls and elastic collisions.
Sr. No Bodies
Action
Reaction
3rd Law of Motion
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Rubber ball
F=1N
F=1N
Action = -Reaction
(elastic)
(5m)
(5m)
Cloth ball
F=1N
F=0.5N
(inelastic)
(5m)
(2.5m)
Flour or mud ball
F=1N
Virtually
(inelastic collision)
(1m)
negligible
Elastic collision
u1
v1
2
3
4
M2 >> M1
Action  -Reaction
Action  Reaction
Action(u1) = Reaction(v1 )
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5
M2 = 1000M1
u1
v1
Action(u1)
=
 -Reaction (v1)
-0.99800199u1
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Typical plastic ball
u1
v1
=
-2u1
Action(u1)
 -Reaction (v1)
(super elastic )
4.0 Generalized form of the Third Law of Motion
It is a basic principle of science that no conclusions can be drawn on the basis of a single or few
qualitative observations. Results are widely accepted only if they can be replicated, and all
possible values of the parameters are taken in account.
On the basis of the above observations The Principia’s Third Law of Motion is generalized:
“To every action there is always an opposed reaction which may or may not be equal in
magnitude, depending upon the inherent characteristics of the process”;
or
“The mutual actions of two bodies may not always be equal, depending upon the inherent
characteristics of the system, and directed in contrary parts.”
Action  Reaction
or Action = -K Reaction
(11)
K is the coefficient of proportionality and depends upon the inherent characteristics of bodies.
The coefficient of proportionality takes into account the inherent characteristics, nature,
composition, flexibility, rigidity, magnitude, distinctiveness of interacting bodies. The bodies
may be of steel, wood, rubber, cloth, wool, sponge, spring, etc. Furthermore, comparative size
and point of impact of target and projectile etc. and roughness of surfaces play significant roles.
In Newton’s third law of motion, there is no term which accounts for above significant factors. It
can be easily justified that the introduction of K in the Third Law of Motion is consistent with
existing concepts in physics. Its value can be determined experimentally. The value of the coefficient of thermal conductivity Z for different materials is shown below. The quantity of heat Q
is determined by the proportionality method.
Q = ZA [T1-T2]t/d
(12)
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The general range of variation of the co-efficient of thermal conductivity Z for
various conductors are:- For Solids: Silver 429 Wm-1K-1, Copper 403 Wm-1K-1, Iron 86
Wm-1K-1, Tin 68.2 Wm-1K-1 Wood 18
Water
Wm-1K-1. For Liquids: Mercury
7.82 Wm-1K-1,
0.599 Wm-1K-1, Acetone 0.167 Wm-1K-1 , C2H5OH 0.226 Wm-1K-1. Likewise the
value of the co-efficient of proportionality K in the Third Law of Motion can be determined.
The universal gravitational constant G is determined experimentally (as it varies so it can be
deemed as constant). Currently the accepted value for the gravitational constant is 6.673 84 ×1011
m3kg-1 s-2, but a recently measured value [5] is much higher, i.e. G = 6.67545 (18)×10-
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m3kg-1s-2. The mass of Earth (M = gR2/G), using the latest value of G, is 0.024% lower.
Thus Newton’s third law of motion establishes universal equality between action and reaction.
Whereas according to the generalized form of the Third Law, action and reaction may or
may not be equal always , depending upon experimental conditions; the value of the
additional coefficient can be determined experimentally. Science is not static.
Acknowledgements
The Author is indebted to various critics; and Professor Robert Bradley, Stephen J. Crothers and
Anjana Sharma for suggestions and discussion.
References
[1] I. Newton, Mathematical Principles of Natural Philosophy (printed for Benjamin Motte,
Middle Temple Gate, London, 1727 ), pp.19-20, translated by Andrew Motte from the Latin.
[2]
http://books.google.co.in/books?id=Tm0FAAAAQAAJ&pg=PA1&redir_esc=y#v=onepage&q&
f=false
[3] http://en.wikipedia.org/wiki/Elastic_collision
[4 ] R. Resnick and D. Halliday, Physics Part I (Wiley Eastern limited, New Delhi, 2nd Ed. 1996
reprinted ), pp.215-22
[5] T. Quinn, H. Parks, C. Speake, and R. Davis, Phys. Rev. Lett. 111, 101102 (2013).
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